Problem: Example

Input

4 4
0 0
6 0
6 6
0 6


Output

35.759506

Solution (in CPP):

Solution:
#include <bits/stdc++.h>

using namespace std;
using uint = unsigned int;

template<class T> using V = vector<T>;

using D = double;
const D PI = acos(D(-1)), EPS = 1e-10;

int sgn(D a) { return (abs(a) <= EPS) ? 0 : (a < 0 ? -1 : 1); }
int sgn(D a, D b) { return sgn(a-b); }
//relative sign
// int rsgn(D a, D f) {
//     if (abs(a) <= f*EPS) return 0;
//     return (a < 0) ? -1 : 1;
// }
struct Pt2 {
    D x, y;
    Pt2(D _x = D(), D _y = D()) : x(_x), y(_y) {}
    Pt2 operator+(const Pt2 &r) const { return Pt2(x+r.x, y+r.y); }
    Pt2 operator-(const Pt2 &r) const { return Pt2(x-r.x, y-r.y); }
    Pt2 operator*(const Pt2 &r) const { return Pt2(x*r.x-y*r.y, x*r.y+y*r.x); }

    Pt2 operator*(const D &r) const { return Pt2(x*r, y*r); }
    Pt2 operator/(const D &r) const { return Pt2(x/r, y/r); }

    Pt2& operator+=(const Pt2 &r) { return *this=*this+r; }
    Pt2& operator-=(const Pt2 &r) { return *this=*this-r; }
    Pt2& operator*=(const Pt2 &r) { return *this=*this*r; }
    Pt2& operator*=(const D &r) { return *this=*this*r; }
    Pt2& operator/=(const D &r) { return *this=*this/r; }
    
    Pt2 operator-() const { return Pt2(-x, -y); }

    bool operator<(const Pt2 &r) const { return 2*sgn(x, r.x)+sgn(y, r.y)<0; }
    bool operator==(const Pt2 &r) const { return sgn((*this-r).rabs()) == 0; }

    D norm() const { return x*x + y*y; }
    D abs() const { return sqrt(norm()); }
    D rabs() const { return max(std::abs(x), std::abs(y)); } // robust abs
    D arg() const { return atan2(y, x); }

    pair<D, D> to_pair() const { return make_pair(x, y); }
    static Pt2 polar(D le, D th) { return Pt2(le*cos(th), le*sin(th)); }
};
ostream& operator<<(ostream& os, const Pt2 &p) {
    return os << "P(" << p.x << ", " << p.y << ")";
}
using P = Pt2;

struct L {
    P s, t;
    L(P _s = P(), P _t = P()) : s(_s), t(_t) {}
    P vec() const { return t-s; }
    D abs() const { return vec().abs(); }
    D arg() const { return vec().arg(); }
};
ostream& operator<<(ostream& os, const L &l) {
    return os << "L(" << l.s << ", " << l.t << ")";
}

D cross(P a, P b) { return a.x*b.y - a.y*b.x; }
D dot(P a, P b) { return a.x*b.x + a.y*b.y; }
// cross(a, b) is too small?
int sgncrs(P a, P b) {
    D cr = cross(a, b);
    if (abs(cr) <= (a.rabs() + b.rabs()) * EPS) return 0;
    return (cr < 0) ? -1 : 1;
}

// -2, -1, 0, 1, 2 : front, clock, on, cclock, back
int ccw(P b, P c) {
    int s = sgncrs(b, c);
    if (s) return s;
    if (!sgn(c.rabs()) || !sgn((c-b).rabs())) return 0;
    if (dot(b, c) < 0) return 2;
    if (dot(-b, c-b) < 0) return -2;
    return 0;
}
int ccw(P a, P b, P c) { return ccw(b-a, c-a); }
int ccw(L l, P p) { return ccw(l.s, l.t, p); }

P project(const L &l, const P &p) {
    P v = l.vec();
    return l.s + v * (dot(v, p-l.s) / v.norm());
}

bool insSL(const L &s, const L &l) {
    int a = ccw(l, s.s), b = ccw(l, s.t);
    return (a%2 == 0 || b%2 == 0 || a != b);
}

bool insSS(const L &s, const L &t) {
    int a = ccw(s, t.s), b = ccw(s, t.t);
    int c = ccw(t, s.s), d = ccw(t, s.t);
    if (a*b <= 0 && c*d <= 0) return true;
    return false;
}

D distLP(const L &l, const P &p) {
return abs(cross(l.vec(), p-l.s)) / l.abs();
}

D distSP(const L &s, const P &p) {
    P q = project(s, p);
    if (ccw(s, q) == 0) return (p - q).abs();
    else return min((s.s - p).abs(), (s.t - p).abs());
}

int crossLL(const L &l, const L &m, P &r) {
    D cr1 = cross(l.vec(), m.vec()), cr2 = cross(l.vec(), l.t - m.s);
    if (sgncrs(l.vec(), m.vec()) == 0) {
        r = l.s;
        if (sgncrs(l.vec(), l.t - m.s)) return 0;
        return -1;
    }
    r = m.s + m.vec() * (cr2 / cr1);
    return 1;
}

using Pol = V<P>;

struct C {
    P p; D r;
    C(P _p = P(), D _r = D()) : p(_p), r(_r) {}
};

//need Intersect/distLP, r.sはよりl.sに近い
int crossCL(const C &c, const L &l, L &r) {
    D u = distLP(l, c.p);
    int si = sgn(u, c.r);    
    if (si == 1) return 0;
    P v = project(l, c.p);
    P di = (si == 0) ? P(0, 0) : l.vec() * (sqrt(c.r*c.r - u*u) / l.abs());
    r = L(v-di, v+di);
    if (si == 0) return 1;
    return 2;
}

//need Intersect/distLP, r.sはよりl.sに近い
int crossCS(const C &c, const L &s, L &l) {
    if (!crossCL(c, s, l)) return 0;
    bool f1 = ccw(s, l.s) == 0, f2 = ccw(s, l.t) == 0;
    if (f1 && f2) return 2;
    if (!f1 && !f2) return 0;
    if (f1) l.t = l.s;
    else l.s = l.t;
    return 1;
}

// C(P(0, 0), r)とTri((0, 0), a, b)の共有面積
D area2CT(const C &c, const P &_a, const P &_b) {
    P a = _a - c.p, b = _b - c.p; D r = c.r;
    if (a == b) return 0;
    auto single = [&](P x, P y, bool tri) {
        if (tri) return cross(x, y);
        else return r * r * ((y * P(x.x, -x.y)).arg());
    };
    bool ia = sgn(a.abs(), r) != 1, ib = sgn(b.abs(), r) != 1;
    if (ia && ib) return single(a, b, true);
//    if (sgn(r, distSP(L(a, b), P(0, 0))) != 1) return single(a, b, false);
    L l;
    if (!crossCS(C(P(0, 0), r), L(a, b), l)) return single(a, b, false);
    if (ia) l.s = l.t;
    else if (ib) l.t = l.s;
    assert(ccw(a, b, l.s) == 0); assert(ccw(a, b, l.t) == 0);
    return single(a, l.s, ia) + single(l.s, l.t, true) + single(l.t, b, ib);
}

// p, cの共有面積
D area2CPol(const C &c, const Pol &po) {
    D sm = 0;
    P a, b = po.back();
    for (auto p: po) {
        a = b; b = p;
        sm += area2CT(c, a, b);
    }
    return sm;
}

int main() {
    cin.tie(0);
    ios::sync_with_stdio(false);
    cout << setprecision(20);
    int n; D r;
    cin >> n >> r;
    Pol pol;
 
    D ymi = 1e100, yma = -1e100;
    for (int i = 0; i < n; i++) {
        D x, y;
        cin >> x >> y;
        pol.push_back(P(x, y));
        ymi = min(ymi, y);
        yma = max(yma, y);
    }
 
    auto calc = [&](D y) {
        D xmi = 1e100, xma = -1e100;
        P a, b = pol.back();
        for (int i = 0; i < n; i++) {
            a = b; b = pol[i];
            P p;
            if (crossLL(L(a, b), L(P(0, y), P(1, y)), p) == 0) continue;
            if (ccw(a, b, p) != 0) continue;
            xmi = min(xmi, p.x); xma = max(xma, p.x);
        }
        D lw = xmi, up = xma;
        for (int ph = 0; ph < 30; ph++) {
            D md1 = (lw+lw+up) / 3;
            D md2 = (lw+up+up) / 3;
            D z1 = area2CPol(C(P(md1, y), r), pol) / 2;
            D z2 = area2CPol(C(P(md2, y), r), pol) / 2;
            if (z1 < z2) {
                lw = md1;
            } else {
                up = md2;
            }
        }
        return area2CPol(C(P(lw, y), r), pol) / 2;
    };
     
    D lw = ymi, up = yma;
    for (int ph = 0; ph < 30; ph++) {
        D md1 = (lw+lw+up) / 3;
        D md2 = (lw+up+up) / 3;
        if (calc(md1) < calc(md2)) {
            lw = md1;
        } else {
            up = md2;
        }
    }
 
    cout << calc(lw) << endl;
    return 0;
}